What is the best tool to monitor mentions on the internet as well as social media?

I eat our own dog food so I might by biased, but you could take Brand24 for a spin. We collect mentions from social media as well as news sites, blogs, forums, and message boards. Historical data is limited though, and the tool works pro-actively so if you want to measure the results of a marketing campaign, for instance, it's best to set up a monitoring project before the launch. You also mentioned cost-effective - there are three pricing plans to choose from, starting from $49 per month. As far as real-time aspect is concerned, it gets even better with a Slack integration (if you use Slack obviously :)).

Very helpful in monitoring all public opinion about the product, not only Facebook, Twitter and Instagram as well as delivers several types of analytical reports which, when used properly, can result in substantial sale boost

Easy to use and very affordable.

Works like a dream with Slack

Syften will provide you with Social News sites monitoring and it integrates with Slack. It has more advanced filters than the alternatives.